New representations for Apéry-like sequences.

*(English)*Zbl 1275.11035Focus of this paper are three sequences satisfying similar difference equations, i.e., the numbers found by R. Apéry [Astérisque 61, 11–13 (1979; Zbl 0401.10049)], the alternating version of the Domb numbers examined by H. H. Chan, S. H. Chan and Z.-G. Liu [Adv. Math. 186, No. 2, 396–410 (2004; Zbl 1122.11087)] and the AZ numbers defined by G. Almkvist and W. Zudilin [Mirror symmetry V. AMS/IP Stud. Adv. Math. 38, 481–515 (2006; Zbl 1118.14043)].

In fact, their generating series \(F_\alpha (z)\), \(F_\delta (z)\) and \(F_\xi (z)\) admit the modular parametrizations via the Hauptmoduls of the three subgroups of index two lying between \(\Gamma_0 (6)\) and its normalizer in \(\mathrm{SL}_2 (\mathbb R)\), studied by H. H. Chan and H. Verrill [Math. Res. Lett. 16, No. 2-3, 405–420 (2009; Zbl 1193.11038)], which are the main tools of the proof, whose secondary means are, e.g., the \(q\)-series expansions of the functions, the differential operator \(\theta =y(d/dy)\) and computer algebra such as Maple.

The authors establish new algebraic relations among the series \(F_\alpha (z)\), \(F_\delta (z)\), \(F_\xi (z)\) and their expressions through \({}_3 F_2\)-hypergeometric series; then they derive three new Ramanujan-type series for \( 1/\pi\) and they auspicate that achievements and methods provided in this paper could help to obtain the explicit evaluation of the three-variable Mahler measures investigated by M. D. Rogers [Ramanujan J. 18, No. 3, 327–340 (2009; Zbl 1226.11113)].

In the proof the authors employ also a congruence by F. Beukers [Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 271–283 (1987; Zbl 0613.10031)], the Kummer’s quadratic transform and the Clausen’s identity both reported in [L. J. Slater, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the simplest Ramanujan’s formula proved by G. Bauer [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)] and the series for \( 1/\pi\) introduced by S. Ramanujan [Q. J. Math., Oxf. 45, 350–372 (1914; JFM 45.1249.01)].

Other Ramanujan-related results supplied by the first-named author {et al.} ([B. C. Berndt, H. H. Chan and S.-S. Huang, Contemp. Math. 254, 79–126 (2000; Zbl 0971.33012)]; B. C. Berndt, H. H. Chan and W.-C. Liaw [J. Number Theory 88, No.1, 129–156 (2001; Zbl 1005.33009)]) and by the second-named author [Modular forms and string duality. Proceedings of a workshop, Banff, Canada, 2006. Fields Institute Communications 54, 179–188 (2008; Zbl 1159.11053)]; [Math. Notes 81, No. 3, 297–301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007; Zbl 1144.33002)] are also recalled in the paper.

In fact, their generating series \(F_\alpha (z)\), \(F_\delta (z)\) and \(F_\xi (z)\) admit the modular parametrizations via the Hauptmoduls of the three subgroups of index two lying between \(\Gamma_0 (6)\) and its normalizer in \(\mathrm{SL}_2 (\mathbb R)\), studied by H. H. Chan and H. Verrill [Math. Res. Lett. 16, No. 2-3, 405–420 (2009; Zbl 1193.11038)], which are the main tools of the proof, whose secondary means are, e.g., the \(q\)-series expansions of the functions, the differential operator \(\theta =y(d/dy)\) and computer algebra such as Maple.

The authors establish new algebraic relations among the series \(F_\alpha (z)\), \(F_\delta (z)\), \(F_\xi (z)\) and their expressions through \({}_3 F_2\)-hypergeometric series; then they derive three new Ramanujan-type series for \( 1/\pi\) and they auspicate that achievements and methods provided in this paper could help to obtain the explicit evaluation of the three-variable Mahler measures investigated by M. D. Rogers [Ramanujan J. 18, No. 3, 327–340 (2009; Zbl 1226.11113)].

In the proof the authors employ also a congruence by F. Beukers [Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 271–283 (1987; Zbl 0613.10031)], the Kummer’s quadratic transform and the Clausen’s identity both reported in [L. J. Slater, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the simplest Ramanujan’s formula proved by G. Bauer [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)] and the series for \( 1/\pi\) introduced by S. Ramanujan [Q. J. Math., Oxf. 45, 350–372 (1914; JFM 45.1249.01)].

Other Ramanujan-related results supplied by the first-named author {et al.} ([B. C. Berndt, H. H. Chan and S.-S. Huang, Contemp. Math. 254, 79–126 (2000; Zbl 0971.33012)]; B. C. Berndt, H. H. Chan and W.-C. Liaw [J. Number Theory 88, No.1, 129–156 (2001; Zbl 1005.33009)]) and by the second-named author [Modular forms and string duality. Proceedings of a workshop, Banff, Canada, 2006. Fields Institute Communications 54, 179–188 (2008; Zbl 1159.11053)]; [Math. Notes 81, No. 3, 297–301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007; Zbl 1144.33002)] are also recalled in the paper.

Reviewer: Enzo Bonacci (Latina)

##### MSC:

11B65 | Binomial coefficients; factorials; \(q\)-identities |

11F11 | Holomorphic modular forms of integral weight |

11F20 | Dedekind eta function, Dedekind sums |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

##### Citations:

Zbl 0401.10049; Zbl 1122.11087; Zbl 1118.14043; Zbl 1193.11038; Zbl 1226.11113; Zbl 0613.10031; Zbl 0135.28101; Zbl 0971.33012; Zbl 1005.33009; Zbl 1159.11053; Zbl 1144.33002; JFM 45.1249.01; ERAM 056.1478cj##### Software:

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\textit{H. H. Chan} and \textit{W. Zudilin}, Mathematika 56, No. 1, 107--117 (2010; Zbl 1275.11035)

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##### References:

[1] | Ramanujan, Q. J. Math. Oxford Ser. (2) 45 pp 350– (1914) |

[2] | Chan, Math. Res. Lett. 16 pp 405– (2009) · Zbl 1193.11038 |

[3] | DOI: 10.1016/j.aim.2003.07.012 · Zbl 1122.11087 |

[4] | DOI: 10.1006/jnth.2000.2615 · Zbl 1005.33009 |

[5] | Berndt, Contemp. Math. 254 pp 79– (2000) |

[6] | DOI: 10.1007/s11139-007-9040-x · Zbl 1226.11113 |

[7] | Apéry, Astérisque 61 pp 11– (1979) |

[8] | Almkvist, Mirror Symmetry V pp 481– (2007) |

[9] | Zudilin, Modular Forms and String Duality (Banff, June 2006) pp 179– (2008) |

[10] | DOI: 10.1134/S0001434607030030 · Zbl 1144.33002 |

[11] | Slater, Generalized Hypergeometric Functions (1966) · Zbl 0135.28101 |

[12] | Bauer, J. für Math. (Crelles J.) 56 pp 101– (1859) |

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